Differential Machine Learning

Axes that matter: PCA with a difference ArXiv ID: 2503.06707 “View on arXiv” Authors: Unknown Abstract We extend the scope of differential machine learning and introduce a new breed of supervised principal component analysis to reduce dimensionality of Derivatives problems. Applications include the specification and calibration of pricing models, the identification of regression features in least-square Monte-Carlo, and the pre-processing of simulated datasets for (differential) machine learning. Keywords: differential machine learning, principal component analysis, derivatives pricing, least-square Monte-Carlo, dimensionality reduction ...

Enforcing asymptotic behavior with DNNs for approximation and regression in finance ArXiv ID: 2411.05257 “View on arXiv” Authors: Unknown Abstract We propose a simple methodology to approximate functions with given asymptotic behavior by specifically constructed terms and an unconstrained deep neural network (DNN). The methodology we describe extends to various asymptotic behaviors and multiple dimensions and is easy to implement. In this work we demonstrate it for linear asymptotic behavior in one-dimensional examples. We apply it to function approximation and regression problems where we measure approximation of only function values (Vanilla Machine Learning''-VML) or also approximation of function and derivative values (Differential Machine Learning’’-DML) on several examples. We see that enforcing given asymptotic behavior leads to better approximation and faster convergence. ...

Mathematics of Differential Machine Learning in Derivative Pricing and Hedging ArXiv ID: 2405.01233 “View on arXiv” Authors: Unknown Abstract This article introduces the groundbreaking concept of the financial differential machine learning algorithm through a rigorous mathematical framework. Diverging from existing literature on financial machine learning, the work highlights the profound implications of theoretical assumptions within financial models on the construction of machine learning algorithms. This endeavour is particularly timely as the finance landscape witnesses a surge in interest towards data-driven models for the valuation and hedging of derivative products. Notably, the predictive capabilities of neural networks have garnered substantial attention in both academic research and practical financial applications. The approach offers a unified theoretical foundation that facilitates comprehensive comparisons, both at a theoretical level and in experimental outcomes. Importantly, this theoretical grounding lends substantial weight to the experimental results, affirming the differential machine learning method’s optimality within the prevailing context. By anchoring the insights in rigorous mathematics, the article bridges the gap between abstract financial concepts and practical algorithmic implementations. ...

Deep Joint Learning valuation of Bermudan Swaptions ArXiv ID: 2404.11257 “View on arXiv” Authors: Unknown Abstract This paper addresses the problem of pricing involved financial derivatives by means of advanced of deep learning techniques. More precisely, we smartly combine several sophisticated neural network-based concepts like differential machine learning, Monte Carlo simulation-like training samples and joint learning to come up with an efficient numerical solution. The application of the latter development represents a novelty in the context of computational finance. We also propose a novel design of interdependent neural networks to price early-exercise products, in this case, Bermudan swaptions. The improvements in efficiency and accuracy provided by the here proposed approach is widely illustrated throughout a range of numerical experiments. Moreover, this novel methodology can be extended to the pricing of other financial derivatives. ...